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Theorem logsqvma2 22677
Description: The Möbius inverse of logsqvma 22676. Equation 10.4.8 of [Shapiro], p. 418. (Contributed by Mario Carneiro, 13-May-2016.)
Assertion
Ref Expression
logsqvma2  |-  ( N  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( ( mmu `  d )  x.  (
( log `  ( N  /  d ) ) ^ 2 ) )  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  d )  x.  (Λ `  ( N  /  d ) ) )  +  ( (Λ `  N )  x.  ( log `  N ) ) ) )
Distinct variable group:    x, d, N

Proof of Theorem logsqvma2
Dummy variables  i 
j  k  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzfid 11779 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
1 ... k )  e. 
Fin )
2 sgmss 22329 . . . . . . . . . 10  |-  ( k  e.  NN  ->  { x  e.  NN  |  x  ||  k }  C_  ( 1 ... k ) )
3 ssfi 7521 . . . . . . . . . 10  |-  ( ( ( 1 ... k
)  e.  Fin  /\  { x  e.  NN  |  x  ||  k }  C_  ( 1 ... k
) )  ->  { x  e.  NN  |  x  ||  k }  e.  Fin )
41, 2, 3syl2anc 654 . . . . . . . . 9  |-  ( k  e.  NN  ->  { x  e.  NN  |  x  ||  k }  e.  Fin )
5 ssrab2 3425 . . . . . . . . . . . 12  |-  { x  e.  NN  |  x  ||  k }  C_  NN
6 simpr 458 . . . . . . . . . . . 12  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  d  e.  { x  e.  NN  |  x  ||  k } )
75, 6sseldi 3342 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  d  e.  NN )
8 vmacl 22341 . . . . . . . . . . 11  |-  ( d  e.  NN  ->  (Λ `  d )  e.  RR )
97, 8syl 16 . . . . . . . . . 10  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (Λ `  d )  e.  RR )
10 dvdsdivcl 22406 . . . . . . . . . . . 12  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
k  /  d )  e.  { x  e.  NN  |  x  ||  k } )
115, 10sseldi 3342 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
k  /  d )  e.  NN )
12 vmacl 22341 . . . . . . . . . . 11  |-  ( ( k  /  d )  e.  NN  ->  (Λ `  ( k  /  d
) )  e.  RR )
1311, 12syl 16 . . . . . . . . . 10  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (Λ `  ( k  /  d
) )  e.  RR )
149, 13remulcld 9402 . . . . . . . . 9  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  e.  RR )
154, 14fsumrecl 13195 . . . . . . . 8  |-  ( k  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d
)  x.  (Λ `  (
k  /  d ) ) )  e.  RR )
16 vmacl 22341 . . . . . . . . 9  |-  ( k  e.  NN  ->  (Λ `  k )  e.  RR )
17 nnrp 10988 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR+ )
1817relogcld 21957 . . . . . . . . 9  |-  ( k  e.  NN  ->  ( log `  k )  e.  RR )
1916, 18remulcld 9402 . . . . . . . 8  |-  ( k  e.  NN  ->  (
(Λ `  k )  x.  ( log `  k
) )  e.  RR )
2015, 19readdcld 9401 . . . . . . 7  |-  ( k  e.  NN  ->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) )  e.  RR )
2120recnd 9400 . . . . . 6  |-  ( k  e.  NN  ->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) )  e.  CC )
2221adantl 463 . . . . 5  |-  ( ( N  e.  NN  /\  k  e.  NN )  ->  ( sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) )  e.  CC )
23 eqid 2433 . . . . 5  |-  ( k  e.  NN  |->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) ) )  =  ( k  e.  NN  |->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) ) )
2422, 23fmptd 5855 . . . 4  |-  ( N  e.  NN  ->  (
k  e.  NN  |->  (
sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) ) : NN --> CC )
25 ssrab2 3425 . . . . . . . . 9  |-  { x  e.  NN  |  x  ||  n }  C_  NN
26 simpr 458 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  n  e.  NN )  /\  m  e.  {
x  e.  NN  |  x  ||  n } )  ->  m  e.  {
x  e.  NN  |  x  ||  n } )
2725, 26sseldi 3342 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  n  e.  NN )  /\  m  e.  {
x  e.  NN  |  x  ||  n } )  ->  m  e.  NN )
28 breq2 4284 . . . . . . . . . . . 12  |-  ( k  =  m  ->  (
x  ||  k  <->  x  ||  m
) )
2928rabbidv 2954 . . . . . . . . . . 11  |-  ( k  =  m  ->  { x  e.  NN  |  x  ||  k }  =  {
x  e.  NN  |  x  ||  m } )
30 oveq1 6087 . . . . . . . . . . . . . 14  |-  ( k  =  m  ->  (
k  /  d )  =  ( m  / 
d ) )
3130fveq2d 5683 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  (Λ `  ( k  /  d
) )  =  (Λ `  ( m  /  d
) ) )
3231oveq2d 6096 . . . . . . . . . . . 12  |-  ( k  =  m  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( m  /  d
) ) ) )
3332adantr 462 . . . . . . . . . . 11  |-  ( ( k  =  m  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( m  /  d
) ) ) )
3429, 33sumeq12dv 13167 . . . . . . . . . 10  |-  ( k  =  m  ->  sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d
)  x.  (Λ `  (
k  /  d ) ) )  =  sum_ d  e.  { x  e.  NN  |  x  ||  m }  ( (Λ `  d )  x.  (Λ `  ( m  /  d
) ) ) )
35 fveq2 5679 . . . . . . . . . . 11  |-  ( k  =  m  ->  (Λ `  k )  =  (Λ `  m ) )
36 fveq2 5679 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( log `  k )  =  ( log `  m
) )
3735, 36oveq12d 6098 . . . . . . . . . 10  |-  ( k  =  m  ->  (
(Λ `  k )  x.  ( log `  k
) )  =  ( (Λ `  m )  x.  ( log `  m
) ) )
3834, 37oveq12d 6098 . . . . . . . . 9  |-  ( k  =  m  ->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) )  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  m } 
( (Λ `  d )  x.  (Λ `  ( m  /  d ) ) )  +  ( (Λ `  m )  x.  ( log `  m ) ) ) )
39 ovex 6105 . . . . . . . . 9  |-  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) )  e.  _V
4038, 23, 39fvmpt3i 5766 . . . . . . . 8  |-  ( m  e.  NN  ->  (
( k  e.  NN  |->  ( sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) ) `  m
)  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  m }  ( (Λ `  d )  x.  (Λ `  ( m  /  d
) ) )  +  ( (Λ `  m
)  x.  ( log `  m ) ) ) )
4127, 40syl 16 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  n  e.  NN )  /\  m  e.  {
x  e.  NN  |  x  ||  n } )  ->  ( ( k  e.  NN  |->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) ) ) `  m )  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  m } 
( (Λ `  d )  x.  (Λ `  ( m  /  d ) ) )  +  ( (Λ `  m )  x.  ( log `  m ) ) ) )
4241sumeq2dv 13164 . . . . . 6  |-  ( ( N  e.  NN  /\  n  e.  NN )  -> 
sum_ m  e.  { x  e.  NN  |  x  ||  n }  ( (
k  e.  NN  |->  (
sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) ) `  m
)  =  sum_ m  e.  { x  e.  NN  |  x  ||  n } 
( sum_ d  e.  {
x  e.  NN  |  x  ||  m }  (
(Λ `  d )  x.  (Λ `  ( m  /  d ) ) )  +  ( (Λ `  m )  x.  ( log `  m ) ) ) )
43 logsqvma 22676 . . . . . . 7  |-  ( n  e.  NN  ->  sum_ m  e.  { x  e.  NN  |  x  ||  n } 
( sum_ d  e.  {
x  e.  NN  |  x  ||  m }  (
(Λ `  d )  x.  (Λ `  ( m  /  d ) ) )  +  ( (Λ `  m )  x.  ( log `  m ) ) )  =  ( ( log `  n ) ^ 2 ) )
4443adantl 463 . . . . . 6  |-  ( ( N  e.  NN  /\  n  e.  NN )  -> 
sum_ m  e.  { x  e.  NN  |  x  ||  n }  ( sum_ d  e.  { x  e.  NN  |  x  ||  m }  ( (Λ `  d )  x.  (Λ `  ( m  /  d
) ) )  +  ( (Λ `  m
)  x.  ( log `  m ) ) )  =  ( ( log `  n ) ^ 2 ) )
4542, 44eqtr2d 2466 . . . . 5  |-  ( ( N  e.  NN  /\  n  e.  NN )  ->  ( ( log `  n
) ^ 2 )  =  sum_ m  e.  {
x  e.  NN  |  x  ||  n }  (
( k  e.  NN  |->  ( sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) ) `  m
) )
4645mpteq2dva 4366 . . . 4  |-  ( N  e.  NN  ->  (
n  e.  NN  |->  ( ( log `  n
) ^ 2 ) )  =  ( n  e.  NN  |->  sum_ m  e.  { x  e.  NN  |  x  ||  n } 
( ( k  e.  NN  |->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d
)  x.  (Λ `  (
k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k
) ) ) ) `
 m ) ) )
4724, 46muinv 22418 . . 3  |-  ( N  e.  NN  ->  (
k  e.  NN  |->  (
sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) )  =  ( i  e.  NN  |->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( (
mmu `  j )  x.  ( ( n  e.  NN  |->  ( ( log `  n ) ^ 2 ) ) `  (
i  /  j ) ) ) ) )
4847fveq1d 5681 . 2  |-  ( N  e.  NN  ->  (
( k  e.  NN  |->  ( sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) ) `  N
)  =  ( ( i  e.  NN  |->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( (
mmu `  j )  x.  ( ( n  e.  NN  |->  ( ( log `  n ) ^ 2 ) ) `  (
i  /  j ) ) ) ) `  N ) )
49 breq2 4284 . . . . . 6  |-  ( k  =  N  ->  (
x  ||  k  <->  x  ||  N
) )
5049rabbidv 2954 . . . . 5  |-  ( k  =  N  ->  { x  e.  NN  |  x  ||  k }  =  {
x  e.  NN  |  x  ||  N } )
51 oveq1 6087 . . . . . . . 8  |-  ( k  =  N  ->  (
k  /  d )  =  ( N  / 
d ) )
5251fveq2d 5683 . . . . . . 7  |-  ( k  =  N  ->  (Λ `  ( k  /  d
) )  =  (Λ `  ( N  /  d
) ) )
5352oveq2d 6096 . . . . . 6  |-  ( k  =  N  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( N  /  d
) ) ) )
5453adantr 462 . . . . 5  |-  ( ( k  =  N  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( N  /  d
) ) ) )
5550, 54sumeq12dv 13167 . . . 4  |-  ( k  =  N  ->  sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d
)  x.  (Λ `  (
k  /  d ) ) )  =  sum_ d  e.  { x  e.  NN  |  x  ||  N }  ( (Λ `  d )  x.  (Λ `  ( N  /  d
) ) ) )
56 fveq2 5679 . . . . 5  |-  ( k  =  N  ->  (Λ `  k )  =  (Λ `  N ) )
57 fveq2 5679 . . . . 5  |-  ( k  =  N  ->  ( log `  k )  =  ( log `  N
) )
5856, 57oveq12d 6098 . . . 4  |-  ( k  =  N  ->  (
(Λ `  k )  x.  ( log `  k
) )  =  ( (Λ `  N )  x.  ( log `  N
) ) )
5955, 58oveq12d 6098 . . 3  |-  ( k  =  N  ->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) )  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  d )  x.  (Λ `  ( N  /  d ) ) )  +  ( (Λ `  N )  x.  ( log `  N ) ) ) )
6059, 23, 39fvmpt3i 5766 . 2  |-  ( N  e.  NN  ->  (
( k  e.  NN  |->  ( sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) ) `  N
)  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  N }  ( (Λ `  d )  x.  (Λ `  ( N  /  d
) ) )  +  ( (Λ `  N
)  x.  ( log `  N ) ) ) )
61 fveq2 5679 . . . . . 6  |-  ( j  =  d  ->  (
mmu `  j )  =  ( mmu `  d ) )
62 oveq2 6088 . . . . . . . 8  |-  ( j  =  d  ->  (
i  /  j )  =  ( i  / 
d ) )
6362fveq2d 5683 . . . . . . 7  |-  ( j  =  d  ->  ( log `  ( i  / 
j ) )  =  ( log `  (
i  /  d ) ) )
6463oveq1d 6095 . . . . . 6  |-  ( j  =  d  ->  (
( log `  (
i  /  j ) ) ^ 2 )  =  ( ( log `  ( i  /  d
) ) ^ 2 ) )
6561, 64oveq12d 6098 . . . . 5  |-  ( j  =  d  ->  (
( mmu `  j
)  x.  ( ( log `  ( i  /  j ) ) ^ 2 ) )  =  ( ( mmu `  d )  x.  (
( log `  (
i  /  d ) ) ^ 2 ) ) )
6665cbvsumv 13157 . . . 4  |-  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  j )  x.  (
( log `  (
i  /  j ) ) ^ 2 ) )  =  sum_ d  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  d )  x.  (
( log `  (
i  /  d ) ) ^ 2 ) )
67 breq2 4284 . . . . . 6  |-  ( i  =  N  ->  (
x  ||  i  <->  x  ||  N
) )
6867rabbidv 2954 . . . . 5  |-  ( i  =  N  ->  { x  e.  NN  |  x  ||  i }  =  {
x  e.  NN  |  x  ||  N } )
69 oveq1 6087 . . . . . . . . 9  |-  ( i  =  N  ->  (
i  /  d )  =  ( N  / 
d ) )
7069fveq2d 5683 . . . . . . . 8  |-  ( i  =  N  ->  ( log `  ( i  / 
d ) )  =  ( log `  ( N  /  d ) ) )
7170oveq1d 6095 . . . . . . 7  |-  ( i  =  N  ->  (
( log `  (
i  /  d ) ) ^ 2 )  =  ( ( log `  ( N  /  d
) ) ^ 2 ) )
7271oveq2d 6096 . . . . . 6  |-  ( i  =  N  ->  (
( mmu `  d
)  x.  ( ( log `  ( i  /  d ) ) ^ 2 ) )  =  ( ( mmu `  d )  x.  (
( log `  ( N  /  d ) ) ^ 2 ) ) )
7372adantr 462 . . . . 5  |-  ( ( i  =  N  /\  d  e.  { x  e.  NN  |  x  ||  i } )  ->  (
( mmu `  d
)  x.  ( ( log `  ( i  /  d ) ) ^ 2 ) )  =  ( ( mmu `  d )  x.  (
( log `  ( N  /  d ) ) ^ 2 ) ) )
7468, 73sumeq12dv 13167 . . . 4  |-  ( i  =  N  ->  sum_ d  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  d )  x.  (
( log `  (
i  /  d ) ) ^ 2 ) )  =  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( ( mmu `  d )  x.  (
( log `  ( N  /  d ) ) ^ 2 ) ) )
7566, 74syl5eq 2477 . . 3  |-  ( i  =  N  ->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  j )  x.  (
( log `  (
i  /  j ) ) ^ 2 ) )  =  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( ( mmu `  d )  x.  (
( log `  ( N  /  d ) ) ^ 2 ) ) )
76 ssrab2 3425 . . . . . . . 8  |-  { x  e.  NN  |  x  ||  i }  C_  NN
77 dvdsdivcl 22406 . . . . . . . 8  |-  ( ( i  e.  NN  /\  j  e.  { x  e.  NN  |  x  ||  i } )  ->  (
i  /  j )  e.  { x  e.  NN  |  x  ||  i } )
7876, 77sseldi 3342 . . . . . . 7  |-  ( ( i  e.  NN  /\  j  e.  { x  e.  NN  |  x  ||  i } )  ->  (
i  /  j )  e.  NN )
79 fveq2 5679 . . . . . . . . 9  |-  ( n  =  ( i  / 
j )  ->  ( log `  n )  =  ( log `  (
i  /  j ) ) )
8079oveq1d 6095 . . . . . . . 8  |-  ( n  =  ( i  / 
j )  ->  (
( log `  n
) ^ 2 )  =  ( ( log `  ( i  /  j
) ) ^ 2 ) )
81 eqid 2433 . . . . . . . 8  |-  ( n  e.  NN  |->  ( ( log `  n ) ^ 2 ) )  =  ( n  e.  NN  |->  ( ( log `  n ) ^ 2 ) )
82 ovex 6105 . . . . . . . 8  |-  ( ( log `  n ) ^ 2 )  e. 
_V
8380, 81, 82fvmpt3i 5766 . . . . . . 7  |-  ( ( i  /  j )  e.  NN  ->  (
( n  e.  NN  |->  ( ( log `  n
) ^ 2 ) ) `  ( i  /  j ) )  =  ( ( log `  ( i  /  j
) ) ^ 2 ) )
8478, 83syl 16 . . . . . 6  |-  ( ( i  e.  NN  /\  j  e.  { x  e.  NN  |  x  ||  i } )  ->  (
( n  e.  NN  |->  ( ( log `  n
) ^ 2 ) ) `  ( i  /  j ) )  =  ( ( log `  ( i  /  j
) ) ^ 2 ) )
8584oveq2d 6096 . . . . 5  |-  ( ( i  e.  NN  /\  j  e.  { x  e.  NN  |  x  ||  i } )  ->  (
( mmu `  j
)  x.  ( ( n  e.  NN  |->  ( ( log `  n
) ^ 2 ) ) `  ( i  /  j ) ) )  =  ( ( mmu `  j )  x.  ( ( log `  ( i  /  j
) ) ^ 2 ) ) )
8685sumeq2dv 13164 . . . 4  |-  ( i  e.  NN  ->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  j )  x.  (
( n  e.  NN  |->  ( ( log `  n
) ^ 2 ) ) `  ( i  /  j ) ) )  =  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  j )  x.  (
( log `  (
i  /  j ) ) ^ 2 ) ) )
8786mpteq2ia 4362 . . 3  |-  ( i  e.  NN  |->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  j )  x.  (
( n  e.  NN  |->  ( ( log `  n
) ^ 2 ) ) `  ( i  /  j ) ) ) )  =  ( i  e.  NN  |->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( (
mmu `  j )  x.  ( ( log `  (
i  /  j ) ) ^ 2 ) ) )
88 sumex 13149 . . 3  |-  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  j )  x.  (
( log `  (
i  /  j ) ) ^ 2 ) )  e.  _V
8975, 87, 88fvmpt3i 5766 . 2  |-  ( N  e.  NN  ->  (
( i  e.  NN  |->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( (
mmu `  j )  x.  ( ( n  e.  NN  |->  ( ( log `  n ) ^ 2 ) ) `  (
i  /  j ) ) ) ) `  N )  =  sum_ d  e.  { x  e.  NN  |  x  ||  N }  ( (
mmu `  d )  x.  ( ( log `  ( N  /  d ) ) ^ 2 ) ) )
9048, 60, 893eqtr3rd 2474 1  |-  ( N  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( ( mmu `  d )  x.  (
( log `  ( N  /  d ) ) ^ 2 ) )  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  d )  x.  (Λ `  ( N  /  d ) ) )  +  ( (Λ `  N )  x.  ( log `  N ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755   {crab 2709    C_ wss 3316   class class class wbr 4280    e. cmpt 4338   ` cfv 5406  (class class class)co 6080   Fincfn 7298   CCcc 9268   RRcr 9269   1c1 9271    + caddc 9273    x. cmul 9275    / cdiv 9981   NNcn 10310   2c2 10359   ...cfz 11424   ^cexp 11849   sum_csu 13147    || cdivides 13518   logclog 21891  Λcvma 22314   mmucmu 22317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-disj 4251  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ioc 11293  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-fac 12036  df-bc 12063  df-hash 12088  df-shft 12540  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-limsup 12933  df-clim 12950  df-rlim 12951  df-sum 13148  df-ef 13336  df-sin 13338  df-cos 13339  df-pi 13341  df-dvds 13519  df-gcd 13674  df-prm 13747  df-pc 13887  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-fbas 17658  df-fg 17659  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cld 18465  df-ntr 18466  df-cls 18467  df-nei 18544  df-lp 18582  df-perf 18583  df-cn 18673  df-cnp 18674  df-haus 18761  df-tx 18977  df-hmeo 19170  df-fil 19261  df-fm 19353  df-flim 19354  df-flf 19355  df-xms 19737  df-ms 19738  df-tms 19739  df-cncf 20296  df-limc 21183  df-dv 21184  df-log 21893  df-vma 22320  df-mu 22323
This theorem is referenced by:  selberg  22682
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